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WORK                                         11
6. In the preceding problem suppose one end of the rod to be fixe while the other end is firmly attached to the middle of another rod perpe dicular to it. A torque of 10 pounds-foot is necessary in order to keep t second rod in a position turned through 15 about the axis of the fii rod. How much work must be done in order to produce an angul deflection of 45 ?
6.   If in problem 5 the torque is due to a couple the forces of which a applied at points 4 inches from the axis of rotation, find the forces appli< and show that the work done by the forces equals the work done by  torque.
7.   Making the following assumption with regard to the normal pressu at the bearings, obtain an expression for the work done in giving a fl wheel an angular displacement.
(Hint.  For this and the following problems consult 57 and 58.)
(a)  Normal pressure is constant.
(b)  Vertical component of the total reaction is constant.
(c)  Normal pressure is a sine function of the angular position;  t latter being measured from the horizontal plane through the axis of t shaft.
(d)  Normal pressure varies as the square of the sine of the angular poi tion.
8.   Find the work done in giving a flywheel a complete rotation.   T following data are given.   The flywheel weighs 5 tons,' the diameter the shaft is 10 inches, the coefficient of friction in the journal bearings 0.05, and the normal pressure on the bearings satisfies one of the folio ing conditions:
(a)  Normal pressure is constant.
(b)  Vertical component of the total reaction is constant.
(c)  Normal pressure varies as the sine of the angular position, measur from the horizontal plane through the axis of the shaft.
(d)  Normal pressure varies as the square of the angular position.
9.   Derive an expression for the work done in giving an angular d placement to a load which is supported by a flat-end pivot.
10.   The rotating parts of a water turbine which weigh 50 tons are su ported by a flat-end pivot.   The diameter of the shaft is 10 inches a: the coefficient of friction is 0.03.   Find the work lost per revolution.
11.   Supposing the normal pressure to be constant derive an expressi for the work done in giving a loaded spherical pivot an angular displa< ment about its axis.
12.   Supposing the normal pressure to be constant derive an expressi for the work done in giving a .loaded conical pivot an angular displaceme]